supxrmnf - Metamath Proof Explorer

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Theorem supxrmnf 12147

Description: Adding minus infinity to a set does not affect its supremum. (Contributed by NM, 19-Jan-2006.)

**Assertion**
Ref	Expression
supxrmnf	⊢ (𝐴 ⊆ ℝ^* → sup((𝐴 ∪ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))

**Proof of Theorem supxrmnf**
Step	Hyp	Ref	Expression
1		uncom 3757	. . 3 ⊢ (𝐴 ∪ {-∞}) = ({-∞} ∪ 𝐴)
2	1	supeq1i 8353	. 2 ⊢ sup((𝐴 ∪ {-∞}), ℝ^, < ) = sup(({-∞} ∪ 𝐴), ℝ^, < )
3		mnfxr 10096	. . . 4 ⊢ -∞ ∈ ℝ^*
4		snssi 4339	. . . 4 ⊢ (-∞ ∈ ℝ^* → {-∞} ⊆ ℝ^*)
5	3, 4	mp1i 13	. . 3 ⊢ (𝐴 ⊆ ℝ^* → {-∞} ⊆ ℝ^*)
6		id 22	. . 3 ⊢ (𝐴 ⊆ ℝ^* → 𝐴 ⊆ ℝ^*)
7		xrltso 11974	. . . . 5 ⊢ < Or ℝ^*
8		supsn 8378	. . . . 5 ⊢ (( < Or ℝ^* ∧ -∞ ∈ ℝ^) → sup({-∞}, ℝ^, < ) = -∞)
9	7, 3, 8	mp2an 708	. . . 4 ⊢ sup({-∞}, ℝ^*, < ) = -∞
10		supxrcl 12145	. . . . 5 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
11		mnfle 11969	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → -∞ ≤ sup(𝐴, ℝ^*, < ))
12	10, 11	syl 17	. . . 4 ⊢ (𝐴 ⊆ ℝ^* → -∞ ≤ sup(𝐴, ℝ^*, < ))
13	9, 12	syl5eqbr 4688	. . 3 ⊢ (𝐴 ⊆ ℝ^* → sup({-∞}, ℝ^, < ) ≤ sup(𝐴, ℝ^, < ))
14		supxrun 12146	. . 3 ⊢ (({-∞} ⊆ ℝ^* ∧ 𝐴 ⊆ ℝ^* ∧ sup({-∞}, ℝ^, < ) ≤ sup(𝐴, ℝ^, < )) → sup(({-∞} ∪ 𝐴), ℝ^, < ) = sup(𝐴, ℝ^, < ))
15	5, 6, 13, 14	syl3anc 1326	. 2 ⊢ (𝐴 ⊆ ℝ^* → sup(({-∞} ∪ 𝐴), ℝ^, < ) = sup(𝐴, ℝ^, < ))
16	2, 15	syl5eq 2668	1 ⊢ (𝐴 ⊆ ℝ^* → sup((𝐴 ∪ {-∞}), ℝ^, < ) = sup(𝐴, ℝ^, < ))

Colors of variables: wff setvar class

Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ∪ cun 3572 ⊆ wss 3574 {csn 4177 class class class wbr 4653 Or wor 5034 supcsup 8346 -∞cmnf 10072 ℝ^*cxr 10073 < clt 10074 ≤ cle 10075

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 ax-cnex 9992 ax-resscn 9993 ax-1cn 9994 ax-icn 9995 ax-addcl 9996 ax-addrcl 9997 ax-mulcl 9998 ax-mulrcl 9999 ax-mulcom 10000 ax-addass 10001 ax-mulass 10002 ax-distr 10003 ax-i2m1 10004 ax-1ne0 10005 ax-1rid 10006 ax-rnegex 10007 ax-rrecex 10008 ax-cnre 10009 ax-pre-lttri 10010 ax-pre-lttrn 10011 ax-pre-ltadd 10012 ax-pre-mulgt0 10013 ax-pre-sup 10014

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-nel 2898 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-ov 6653 df-oprab 6654 df-mpt2 6655 df-er 7742 df-en 7956 df-dom 7957 df-sdom 7958 df-sup 8348 df-pnf 10076 df-mnf 10077 df-xr 10078 df-ltxr 10079 df-le 10080 df-sub 10268 df-neg 10269

This theorem is referenced by: supxrmnf2 39660

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